Multidimensional Generalization

For the multidimensional generalization of Eqs. (3.20)-(3.30), we consider the F—dimensional Hamiltonian operator given by 

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Hershkovitz E and Poliak E 1997 Multidimensional generalization of the PGH turnover theory for activated rate processes J. Chem. Phys. 106 7678... 

This theorem is a multidimensional generalization of the geometric arguments given previously. By result 1, in searching for a solution, we need only look at vertices. It is thus of interest to know how to characterize vertices in many dimensions algebraically. This information is given by the next result. 

If the PCET reaction involves electronically adiabatic ET and PT and is vibrationally adiabatic, the system moves on the lowest two-dimensional energy surface equivalent to the lowest eigenvalue of the matrix H. In this case, the rate could be calculated with the multidimensional generalization of the Grote-Hynes theory [29-31]. 

The paper is organized as follows. In Sec. 1, we introduce the main features of quantum error-correction, and, particularly, we present the already well-developed theory of quantum error-correcting codes. In Sec. 2, we present a multidimensional generalization of the quantum Zeno effect and its application to the protection of the information contained in compound systems. Moreover, we suggest a universal physical implementation of the coding and decoding steps through the non-holonomic control. Finally, in Sec. 3, we focus on the application of our method to a rubidium isotope. 

The section starts by the presentation of a multidimensional generalization of the quantum Zeno effect, which we then employ to protect information in compound systems. Finally, we suggest the non-holonomic control technique as a physical way to implement the coding / decoding steps of our scheme. 

The multidimensional generalization of the quantum Zeno effect we have just described allows us to protect an arbitrary subspace C of a Hilbert space H... 

This is the multidimensional generalization of the position autocorrelation function of a Brownian oscillator and provides such a function with a number of unknown parameters. Using this approximate memory function allows replacement of the GLE in Eq. (5.1) by... 

Kramers model is simplistic. It is one-dimensional and assumes that the friction is Markovian—uncorrelated in time. A multidimensional generalization of Kramers problem in the spatial diffusion limit was proposed and solved by Langer (18). The multidimensional energy diffusion limit was solved by Matkowsky, Schuss, and coworkers (19,20). A multidimensional turnover theory has been recently formulated (21). 

The theory for the depopulation factor Y for the GLE and the STGLE is discussed in Sec. VI. Multidimensional generalizations for the depopulation and the spatial diffusion factors are presented in Sec. VII. Extension of the theory to include motion on periodic potentials and surface diffusion is given in Sec. VIII. We end with a discussion of future directions and topics which remain unsolved at this point. 

The separability of the Hamiltonian in the normal mode form implies that the dynamics is in some sense trivial. One must only consider the continuum limit of a collection of independent harmonic oscillators and a single parabolic barrier. As described in Sec. III.D, this simple dynamics leads to some important relations between the Hamiltonian approach and the more standard stochastic theories. Multidimensional generalization of the parabolic barrier case will be discussed briefly in Sec. VIII. 

The surface t v-1 is a particular example of what Wiggins has referred to as a hyperbolic manifold and what De Leon and Ling have termed a normally invariant hyperbolic manifoldd Hyperbolic manifolds are unstable and constitute the formal multidimensional generalization of unstable periodic orbits. Hyperbolic manifolds, like PODS, can be either repulsive or attractive. - If motion near a hyperbolic manifold falls away without recrossing it in configuration space, the hyperbolic manifold is said to be repulsive. On the other hand, it is often the case that motion near a hyperbolic manifold will cross it several times in configuration space as it falls away, and in this case it is said to be attractive. 

We now give the sinc-function based DVR of the power series Green s function. For simplicity, we restrict our attention to a one-dimensional system. The multidimensional generalization is straightforward, and will be given afterwards. Letting... 

We consider a one dimensional radial system with coordinate R, The multidimensional generalization is extremely straightforward. The time independent Schrodinger equation for a scattering system at energy E with outgoing waves (denoted by 4- ) in all open channels and an incoming wave in reactant channel iir is... 

The multidimensional generalization of these equations is straightforward. Expectation values are readily obtained fi-om Eq. (15) and these expressions. Similarly, if the operator A is given by a low-order polynomial, the coherent state transform of p A can be evaluated analytically. 

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